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Homework answers / question archive / Problem Set 3 Name Due: Friday, April 2, 2021 Solve each of the following problems

Problem Set 3 Name Due: Friday, April 2, 2021 Solve each of the following problems. For any series solutions you obtain, you need only give the recursion formula and write out the …rst three non-zero terms of the series. Do not write on this sheet but attach it to the top as a cover sheet. 1. Brie‡y explain why x0 = 0 is an ordinary point of the di¤erential equation (4 x2 )y 00 xy 0 4y = 0 and, without solving the equation, …nd an interval on which any series solution would converge. Be sure to justify your conclusion. Find two linearly independent solutions y1 (x) and y2 (x) where (i) y1 (0) = 1; y10 (0) = 0 and (ii) y2 (0) = 0; y20 (0) = 1: 2. The ODE x3 y 00 + (x2 x3 )y 0 (4 sin x)y = 0. Find the singular point and give a brief reason why it is a singular point, classify the singular point as regular or irregular and be sure to justify your conclusion. If the singular point is a regular singular point then, without di¤erentiating any series, (i) …nd the indicial equation and the exponents m1 m2 , (ii) determine the smallest interval on which any Frobenius series solution must converge, (iii) one of these exponents will always give a Frobenius series solution, which one?, and (iv) will the other exponent give a Frobenius series solution? Yes? NO? Maybe? Brie‡y explain. 3. Consider the di¤erential equation x3 y 00 + xy 0 y = 0. (a) Show that x0 = 0 is an irregular singular point. (b) Find two linearly independent solutions by guessing one and building a second solution. (c) Show that this second solution can not be expressed as a Frobenius series. 4. For each of the following di¤erential equations; (i) show that there is a regular singular point at x0 = 0, (ii) determine the interval on which any Frobenius series solution must hold, (iii) …nd two linearly independent Frobenius series solutions if possible, if not, …nd one and show why you can not get a second one. Be sure to give the recursion formulas and exponents (a) 3xy 00 + (6x 2)y 0 + 2y = 0 (b) x2 y 00 + xy 0 + (x4 4)y = 0 5. For the Bessel equation with p = 21 ; x2 y 00 +xy 0 +(x2 and J y1 = 1=2 (x) cos px x and (a) Hence p = 1 X 1 4) = 0 we know that J1=2 (x) = 1 X ( x )2n+1=2 ( 1)n n!2 (3=2+n) n=0 ( x )2n 2 ( 1)n n! n=0 px y2 = sin x 1=2 (1=2+n) are 2 linearly independent solutions. It can also be shown that are linearly independent solutions of this Bessel equation. xJ1=2 (x) = a cos x + b sin x and q 2 to show that J1=2 (x) = x sin x and J p xJ 1=2 (x) 1=2 (x) = q = c cos x + d sin x: Evaluate these constants 2 x cos x (b) Establish these formulas by direct manipulation of the series expansions of J1=2 (x) and J 1 1=2 (x): Problem Set 2 Due: March 8, 2021 Name Do each of the following problems. Be sure to show all your work. You are not allowed to discuss this assignment with any living being other than the instructor. You may use your notes and textbook but may not use any other source. You are not allowed to use any formulas for methods of solving di¤erential equations. Do not write on this sheet but use it as a cover sheet. 1. Given the general homogeneous equation y 00 + P (x) y 0 + Q(x)y = 0 show that if y1 (x) and y2 (x) are two solutions of this homogeneous equation on the interval [a; b] and have a common zero on this interval, then y1 (x) and y2 (x) are linearly dependent. 2. Consider the two functions f (x) = x3 and g(x) = x2 jxj on the interval [ 1; 1]. (a) Show that their Wronskian W (f; g) vanishes identically on the interval [ 1; 1]. (b) Show that f and g are not linearly dependent on the interval [ 1; 1]. (c) Do a: and b: contradict Lemma 2 (page 116 of the text). Brie‡y explain your conclusion. 3. Solve each of the following di¤erential equations: (a) y 00 6y 0 + 9y = e3x (b) y 00 + 4y 0 + 8y = 6 cos x (c) y iv 4y 00 = 22 sin x 96x2 + 24x + 112 (d) x2 y 00 + 3xy 0 + 2y = 0; y(1) = 2; y 0 (1) = 0 (e) x2 y 00 (f) (x2 xy 0 + y = 2 + 4 ln x, y(1) = 8; y 0 (1) = 8 1)y 00 2xy 0 + 2y = (x2 1)2 1 Math 254 A Di¤erential Equations Spring 2021 Instructor: Fred Schultheis Class meetings and location: MWF, 1:00-2:10 in PPHAC 335 and online Final Exam: Wednesday May 5, 2021 at 10:15 am O¢ ce: SMC 342 O¢ ce Hours: Monday 11:00-12:00, Thursday 9:00-10:00, and by appointment Phone: 610-625-7887 Required Text: Di¤erential Equations with Applications and Historical Notes, Third Edition (Textbooks in Mathematics), George Simmons (You can download a free PDF of this text at https://www.academia.edu/40299979/Di¤erential_Equations_with_applications_3_Ed_George Email: schultheisf@moravian.edu Course Prerequisite: Math 211 To make an appointment email me 2 or 3 times when you can meet on ZOOM. I teach MWF 8:00-9:10, 9:40-10:50, and 1:00-2:10.and am usually available at most other times. This course is a natural continuation of the 3 semester calculus sequence in which one applies most of the concepts of calculus. It is important that you have a very …rm grasp of ordinary and partial di¤erentiation, basic techniques of integration, and power series. The main content of the course is contained in Chapters 1, 2, 4, and 8 of the text. Other topics will be covered as time permits. Course Description Homework assignments will be given at each class meeting. Students are expected to complete these assignments by the next class meeting, where they will be discussed. No one can learn mathematics without doing it themselves and so, to the student, homework is the most important part of the course. In addition to the daily homework assignments (ungraded) there will be regular problem sets (graded). Since class participation is important, students are expected to attend every class. Coarse Goals Upon completing the course, successful students will be able to identify and classify the various types of ordinary di¤erential equations, be pro…cient in the various techniques needed to solve several di¤erent types of di¤erential equations, understand ordinary di¤erential equations conceptually and be able to use them to model problems, and gain a further appreciation of the beauty and utility of mathematics. Grading Your …nal grade will be based on; 3 graded problem sets (100 points each), 3 equally weighted hourly exams (100 points each), and a comprehensive …nal exam (at most one third of your total grade). Graded assignments must be turned in by the beginning of class on the date due to be graded without penalty. No assignment will be accepted after graded papers have been returned to the students. The following grading scale is used for assigning your …nal grade. 93 100 A 90 92 A 87 83 80 89 B+ 86 B 82 B 77 73 70 1 79 C+ 76 C 72 C 67 63 60 69 D+ 66 D 62 D 59 F Attendance Class attendance is strongly encouraged. You are responsible for all work covered in class and all assignments, even if absent from class. If you must miss more than one class due to illness or emergency, you should notify the instructor. In-class exams must be taken at the announced time; make-up exams will be given only in case of extreme emergency or serious illness. Extra Credit One goal for this course is to develop an appreciation of the beauty and utility of mathematics. To help foster this appreciation you should spend some time outside of class thinking and discussing mathematics. There are no speci…c requirements for this portion of the course but many opportunities for you to earn extra credit. Some examples of such activities include: attending talks, giving a talk, reading a paper, or solving a problem outside the scope of the class (must be preapproved by the instructor): If you attend an event relevant to your mathematical growth, then to earn extra credit you need to write a short paper that explains what the event was and how it deepened your appreciation of di¤erential equations or mathematics. For any talks you attend a write up is due within one week of when the talk was given. All write ups should include the date and title of the talk and the name of the speaker. No extra credit points will be accepted after the second last Friday (April 23, 2021) of the term. Learning Disability Accommodations “Moravian College is committed to ensuring the full participation of all students in its programs. If you have a documented disability (or think you may have a disability) and, as a result, need a reasonable accommodation to participate in this class, complete course requirements, or bene…t from the College’s programs or services, contact the Accessibility Services Center (ASC) as soon as possible. To receive any academic accommodation, you must be appropriately registered with ASC. The ASC works with students con…dentially and does not disclose any disability-related information without their permission. To contact the Accessibility Services Center (ASC), located in the lower level of Monocacy Hall, stop in, call 610-861-1401 or email: asc@moravian.edu.” Academic Support Moravian o¤ers a variety of kinds of support for academics beyond the classroom. For more information on the o¤erings that can help you be successful in classes, visit the Academic Support website (https://www.moravian.edu/academic-support) All academic support offerings are free to all Moravian students. Academic Support includes peer learning and tutoring of several kinds. All tutoring begins in the second week of classes. At that time, drop-in tutoring services become available for a variety of di¤erent subjects and courses. For a full list of these for each semester, visit this website. These sessions do not require an appointment. For subjects and classes not covered by drop-in tutoring, peer tutoring may still be available. Please read more about tutoring and peer learning options, policies, and operations here. If you have questions about tutoring or other academic support services after reviewing the information in these links, please contact Mr. Matthew Werkheiser, Coordinator for Tutoring and Academic Support, at 610-625-7843 or by email at werkheiserm@moravian.edu. 2 ACADEMIC HONESTY POLICY GUIDELINES-MATHEMATICS COURSES The Mathematics and Computer Science Department supports and is governed by the Academic Honesty Policy of Moravian College as stated in the Moravian College Student Handbook, also available on AMOS. The following statements will help clarify the policies of members of the Mathematics faculty. In all homework assignments which are to be graded, you may use your class notes and any books or library sources. When you use the ideas or thoughts of others, however, you must acknowledge the source. For graded homework assignments, you may not use a solution manual or the help, orally or in written form, of an individual other than your instructor. If you receive help from anyone other than your instructor or if you fail to reference your sources you will be violating the Academic Honesty Policy of Moravian College. For homework which is not to be graded, if you choose, you may work with your fellow students. You are responsible for understanding and being able to explain the solution of all assigned problems, both graded and ungraded. Title IX "Moravian College faculty are committed to providing a learning environment free from harassment and discrimination, including sexual harassment/violence under Title IX. Should a student disclose a concern of this nature, the faculty member is obligated to inform the Title IX Coordinator, who will assist the student in determining support measures and resolution options. Reports can be made online anytime at www.moravian.edu/titleix. Fully con…dential reporting options include the College Chaplains and professionals in the Counseling and Health Centers. Survivors are encouraged to seek immediate assistance by contacting the Advocates at (484) 764-9242. For more information, including grievance procedures, please view the Equal Opportunity, Harassment, and Non-Discrimination Policy at www.moravian.edu/policy. Note: This syllabus is a guideline for the course. It may be necessary to make changes during the semester. I will announce any changes in class. 3 DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES Third Edition TEXTBOOKS in MATHEMATICS Series Editors: Al Boggess and Ken Rosen PUBLISHED TITLES ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH, SECOND EDITION William Paulsen ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom ADVANCED LINEAR ALGEBRA Hugo Woerdeman APPLIED ABSTRACT ALGEBRA WITH MAPLE™ AND MATLAB®, THIRD EDITION Richard Klima, Neil Sigmon, and Ernest Stitzinger APPLIED DIFFERENTIAL EQUATIONS: THE PRIMARY COURSE Vladimir Dobrushkin COMPUTATIONAL MATHEMATICS: MODELS, METHODS, AND ANALYSIS WITH MATLAB® AND MPI, SECOND EDITION Robert E. White DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION Steven G. Krantz DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE WITH BOUNDARY VALUE PROBLEMS Steven G. Krantz DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY Mark A. McKibben and Micah D. Webster ELEMENTARY NUMBER THEORY James S. Kraft and Lawrence C. Washington EXPLORING LINEAR ALGEBRA: LABS AND PROJECTS WITH MATHEMATICA® Crista Arangala GRAPHS & DIGRAPHS, SIXTH EDITION Gary Chartrand, Linda Lesniak, and Ping Zhang INTRODUCTION TO ABSTRACT ALGEBRA, SECOND EDITION Jonathan D. H. Smith TEXTBOOKS in MATHEMATICS DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES Third Edition George F. Simmons CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20160815 International Standard Book Number-13: 978-1-4987-0259-1 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com For Hope and Nancy my wife and daughter who still make it all worthwhile Contents Preface to the Third Edition .................................................................................xi Preface to the Second Edition ............................................................................ xiii Preface to the First Edition...................................................................................xv Suggestions for the Instructor ........................................................................... xix About the Author ................................................................................................ xxi 1. The Nature of Differential Equations. Separable Equations ................1 1 Introduction ...........................................................................................1 2 General Remarks on Solutions ............................................................4 3 Families of Curves. Orthogonal Trajectories .................................. 11 4 Growth, Decay, Chemical Reactions, and Mixing.......................... 19 5 Falling Bodies and Other Motion Problems ................................... 31 6 The Brachistochrone. Fermat and the Bernoullis ........................... 40 Appendix A: Some Ideas From the Theory of Probability: The Normal Distribution Curve (or Bell Curve) and Its Differential Equation ................................................................................ 51 2. First Order Equations ..................................................................................65 7 Homogeneous Equations ...................................................................65 8 Exact Equations ................................................................................... 69 9 Integrating Factors .............................................................................. 74 10 Linear Equations ................................................................................. 81 11 Reduction of Order .............................................................................85 12 The Hanging Chain. Pursuit Curves ............................................... 88 13 Simple Electric Circuits ...................................................................... 95 3. Second Order Linear Equations............................................................... 107 14 Introduction ....................................................................................... 107 15 The General Solution of the Homogeneous Equation ................. 113 16 The Use of a Known Solution to find Another ............................. 119 17 The Homogeneous Equation with Constant Coefficients ........... 122 18 The Method of Undetermined Coefficients .................................. 127 19 The Method of Variation of Parameters......................................... 133 20 Vibrations in Mechanical and Electrical Systems ........................ 136 21 Newton’s Law of Gravitation and The Motion of the Planets ...... 146 22 Higher Order Linear Equations. Coupled Harmonic Oscillators ....................................................................... 155 23 Operator Methods for Finding Particular Solutions.................... 161 Appendix A. Euler ....................................................................................... 170 Appendix B. Newton ................................................................................... 179 vii viii Contents 4. Qualitative Properties of Solutions ........................................................ 187 24 Oscillations and the Sturm Separation Theorem ......................... 187 25 The Sturm Comparison Theorem ................................................... 194 5. Power Series Solutions and Special Functions..................................... 197 26 Introduction. A Review of Power Series ........................................ 197 27 Series Solutions of First Order Equations ...................................... 206 28 Second Order Linear Equations. Ordinary Points ....................... 210 29 Regular Singular Points ................................................................... 219 30 Regular Singular Points (Continued) ............................................. 229 31 Gauss’s Hypergeometric Equation ................................................. 236 32 The Point at Infinity .......................................................................... 242 Appendix A. Two Convergence Proofs .................................................... 246 Appendix B. Hermite Polynomials and Quantum Mechanics.............. 250 Appendix C. Gauss ...................................................................................... 262 Appendix D. Chebyshev Polynomials and the Minimax Property ...... 270 Appendix E. Riemann’s Equation ............................................................. 278 6. Fourier Series and Orthogonal Functions ............................................. 289 33 The Fourier Coefficients ................................................................... 289 34 The Problem of Convergence .......................................................... 301 35 Even and Odd Functions. Cosine and Sine Series ....................... 310 36 Extension to Arbitrary Intervals ..................................................... 319 37 Orthogonal Functions ...................................................................... 325 38 The Mean Convergence of Fourier Series ...................................... 336 Appendix A. A Pointwise Convergence Theorem ..................................345 7. Partial Differential Equations and Boundary Value Problems ........ 351 39 Introduction. Historical Remarks ................................................... 351 40 Eigenvalues, Eigenfunctions, and the Vibrating String .............. 355 41 The Heat Equation ............................................................................ 366 42 The Dirichlet Problem for a Circle. Poisson’s Integral ................. 372 43 Sturm–Liouville Problems ............................................................... 379 Appendix A. The Existence of Eigenvalues and Eigenfunctions .......... 388 8. Some Special Functions of Mathematical Physics .............................. 393 44 Legendre Polynomials ...................................................................... 393 45 Properties of Legendre Polynomials ..............................................400 46 Bessel Functions. The Gamma Function ....................................... 407 47 Properties of Bessel Functions ........................................................ 418 Appendix A. Legendre Polynomials and Potential Theory................... 427 Appendix B. Bessel Functions and the Vibrating Membrane................ 435 Appendix C. Additional Properties of Bessel Functions........................ 441 Contents ix 9. Laplace Transforms .................................................................................... 447 48 Introduction ....................................................................................... 447 49 A Few Remarks on the Theory ....................................................... 452 50 Applications to Differential Equations .......................................... 457 51 Derivatives and Integrals of Laplace Transforms ........................463 52 Convolutions and Abel’s Mechanical Problem ............................. 468 53 More about Convolutions. The Unit Step and Impulse Functions ............................................................................ 475 Appendix A. Laplace ...................................................................................483 Appendix B. Abel .........................................................................................484 10. Systems of First Order Equations............................................................ 487 54 General Remarks on Systems .......................................................... 487 55 Linear Systems................................................................................... 491 56 Homogeneous Linear Systems with Constant Coefficients ........ 498 57 Nonlinear Systems. Volterra’s Prey-Predator Equations ............. 507 11. Nonlinear Equations .................................................................................. 513 58 Autonomous Systems. The Phase Plane and Its Phenomena ..... 513 59 Types of Critical Points. Stability .................................................... 519 60 Critical Points and Stability for Linear Systems ........................... 529 61 Stability By Liapunov’s Direct Method.......................................... 541 62 Simple Critical Points of Nonlinear Systems ................................ 547 63 Nonlinear Mechanics. Conservative Systems............................... 557 64 Periodic Solutions. The Poincaré–Bendixson Theorem............... 563 65 More about the van der Pol Equation............................................. 572 Appendix A. Poincaré ................................................................................. 574 Appendix B. Proof of Liénard’s Theorem ................................................ 576 12. The Calculus of Variations ....................................................................... 581 66 Introduction. Some Typical Problems of the Subject ................... 581 67 Euler’s Differential Equation for an Extremal .............................. 584 68 Isoperimetric Problems .................................................................... 595 Appendix A. Lagrange ................................................................................ 606 Appendix B. Hamilton’s Principle and Its Implications ........................608 13. The Existence and Uniqueness of Solutions ......................................... 621 69 The Method of Successive Approximations .................................. 621 70 Picard’s Theorem ............................................................................... 626 71 Systems. The Second Order Linear Equation ............................... 638 14. Numerical Methods ...................................................................................643 By John S. Robertson 72 Introduction .......................................................................................643 x Contents 73 74 75 76 77 The Method of Euler .........................................................................646 Errors ..................................................................................................650 An Improvement to Euler ................................................................ 652 Higher Order Methods..................................................................... 657 Systems ............................................................................................... 661 Numerical Tables ............................................................................................... 667 Answers ............................................................................................................... 681 Index ..................................................................................................................... 723 Preface to the Third Edition I have taken advantage of this new edition of my book on differential equations to add two batches of new material of independent interest: First, a fairly substantial appendix at the end of Chapter 1 on the famous bell curve. This curve is the graph of the normal distribution function, with many applications in the natural sciences, the social sciences, mathematics—in statistics and probability theory—and engineering. We shall be especially interested how the differential equation for this curve arises from very simple considerations and can be solved to obtain the equation of the curve itself. And second, a brief section on the van der Pol nonlinear equation and its historical background in World War II that gave it significance in the development of the theory of radar. This consists, in part, of personal recollections of the eminent physicist Freeman Dyson. Finally, I should add a few words on the meaning of the cover design, for this design amounts to a bit of self-indulgence. The chapter on Fourier series is there mainly to provide machinery needed for the following chapter on partial differential equations. However, one of the minor offshoots of Fourier series is to find the exact sum of the infinite series formed from the reciprocals of the squares of the positive integers (the first formula on the cover). This sum was discovered by the great Swiss mathematician Euler in 1736, and since his time, several other methods for obtaining this sum, in addition to his own, have been discovered. This is one of the topics dealt with in Sections 34 and 35 and has been one of my own minor hobbies in mathematics for many years. However, from 1736 to the present day, no one has ever been able to find the exact sum of the reciprocals of the cubes of the positive integers (the second formula on the cover). Some years ago, I was working with the zeroes of the Bessel functions. I thought for an exciting period of several days that I was on the trail of this unknown sum, but in the end it did not work out. Instead, the trail deviated in an unexpected direction and yielded yet another method for finding the sum in the first formula. These ideas will be found in Section 47. xi Preface to the Second Edition “As correct as a second edition”—so goes the idiom. I certainly hope so, and I also hope that anyone who detects an error will do me the kindness of letting me know, so that repairs can be made. As Confucius said, “A man who makes a mistake and doesn’t correct it is making two mistakes.” I now understand why second editions of textbooks are always longer than first editions: as with governments and their budgets, there is always strong pressure from lobbyists to put things in, but rarely pressure to take things out. The main changes in this new edition are as follows: the number of problems in the first part of the book has been more than doubled; there are two new chapters, on Fourier Series and on Partial Differential Equations; sections on higher order linear equations and operator methods have been added to Chapter 3; and further material on convolutions and engineering applications has been added to the chapter on Laplace Transforms. Altogether, many different one-semester courses can be built on various parts of this book by using the schematic outline of the chapters given on page xix. There is even enough material here for a two-semester course, if the appendices are taken into account. Finally, an entirely new chapter on Numerical Methods (Chapter 14) has been written especially for this edition by Major John S. Robertson of the United States Military Academy. Major Robertson’s expertise in these matters is much greater than my own, and I am sure that many users of this new edition will appreciate his contribution, as I do. McGraw-Hill and I would like to thank the following reviewers for their many helpful comments and suggestions: D. R. Arterburn, New Mexico Tech; Edward Beckenstein, St. John’s University; Harold Carda, South Dakota School of Mines and Technology; Wenxiong Chen, University of Arizona; Jerald P. Dauer, University of Tennessee; Lester B. Fuller, Rochester Institute of Technology; Juan Gatica, University of Iowa; Richard H. Herman, The Pennsylvania State University; Roger H. Marty, Cleveland State University; Jean-Pierre Meyer, The Johns Hopkins University; Krzysztof Ostaszewski, University of Louisville; James L. Rovnyak, University of Virginia; Alan Sharples, New Mexico Tech; Bernard Shiffman, The Johns Hopkins University; and Calvin H. Wilcox, University of Utah. George F. Simmons xiii Preface to the First Edition To be worthy of serious attention, a new textbook on an old subject should embody a definite and reasonable point of view which is not represented by books already in print. Such a point of view inevitably reflects the experience, taste, and biases of the author, and should therefore be clearly stated at the beginning so that those who disagree can seek nourishment elsewhere. The structure and contents of this book express my personal opinions in a variety of ways, as follows. The place of differential equations in mathematics. Analysis has been the dominant branch of mathematics for 300 years, and differential equations are the heart of analysis. This subject is the natural goal of elementary calculus and the most important part of mathematics for understanding the physical sciences. Also, in the deeper questions it generates, it is the source of most of the ideas and theories which constitute higher analysis. Power series, Fourier series, the gamma function and other special functions, integral equations, existence theorems, the need for rigorous justifications of many analytic processes—all these themes arise in our work in their most natural context. And at a later stage they provide the principal motivation behind complex analysis, the theory of Fourier series and more general orthogonal expansions, Lebesgue integration, metric spaces and Hilbert spaces, and a host of other beautiful topics in modern mathematics. I would argue, for example, that one of the main ideas of complex analysis is the liberation of power series from the confining environment of the real number system; and this motive is most clearly felt by those who have tried to use real power series to solve differential equations. In botany, it is obvious that no one can fully appreciate the blossoms of flowering plants without a reasonable understanding of the roots, stems, and leaves which nourish and support them. The same principle is true in mathematics, but is often neglected or forgotten. Fads are as common in mathematics as in any other human activity, and it is always difficult to separate the enduring from the ephemeral in the achievements of one’s own time. At present there is a strong current of abstraction flowing through our graduate schools of mathematics. This current has scoured away many of the individual features of the landscape and replaced them with the smooth, rounded boulders of general theories. When taken in moderation, these general theories are both useful and satisfying; but one unfortunate effect of their predominance is that if a student doesn’t learn a little while he is an undergraduate about such colorful and worthwhile topics as the wave equation, Gauss’s hypergeometric function, the gamma function, and the basic problems of the calculus of xv xvi Preface to the First Edition variations—among many others—then he is unlikely to do so later. The natural place for an informal acquaintance with such ideas is a leisurely introductory course on differential equations. Some of our current books on this subject remind me of a sightseeing bus whose driver is so obsessed with speeding along to meet a schedule that his passengers have little or no opportunity to enjoy the scenery. Let us be late occasionally, and take greater pleasure in the journey. Applications. It is a truism that nothing is permanent except change; and the primary purpose of differential equations is to serve as a tool for the study of change in the physical world. A general book on the subject without a reasonable account of its scientific applications would therefore be as futile and pointless as a treatise on eggs that did not mention their reproductive purpose. This book is constructed so that each chapter except the last has at least one major “payoff”—and often several—in the form of a classic scientific problem which the methods of that chapter render accessible. These applications include The brachistochrone problem The Einstein formula E = mc2 Newton’s law of gravitation The wave equation for the vibrating string The harmonic oscillator in quantum mechanics Potential theory The wave equation for the vibrating membrane The prey–predator equations Nonlinear mechanics Hamilton’s principle Abel’s mechanical problem I consider the mathematical treatment of these problems to be among the chief glories of Western civilization, and I hope the reader will agree. The problem of mathematical rigor. On the heights of pure mathematics, any argument that purports to be a proof must be capable of withstanding the severest criticisms of skeptical experts. This is one of the rules of the game, and if you wish to play you must abide by the rules. But this is not the only game in town. There are some parts of mathematics—perhaps number theory and abstract algebra—in which high standards of rigorous proof may be appropriate at all levels. But in elementary differential equations a narrow insistence on doctrinaire exactitude tends to squeeze the juice out of the subject, so that only the dry husk remains. My main purpose in this book is to help the Preface to the First Edition xvii student grasp the nature and significance of differential equations; and to this end, I much prefer being occasionally imprecise but understandable to being completely accurate but incomprehensible. I am not at all interested in building a logically impeccable mathematical structure, in which definitions, theorems, and rigorous proofs are welded together into a formidable barrier which the reader is challenged to penetrate. In spite of these disclaimers, I do attempt a fairly rigorous discussion from time to time, notably in Chapter 13 and Appendices A in Chapters 5, 6 and 7, and B in Chapter 11. I am not saying that the rest of this book is nonrigorous, but only that it leans toward the activist school of mathematics, whose primary aim is to develop methods for solving scientific problems—in contrast to the contemplative school, which analyzes and organizes the ideas and tools generated by the activists. Some will think that a mathematical argument either is a proof or is not a proof. In the context of elementary analysis I disagree, and believe instead that the proper role of a proof is to carry reasonable conviction to one’s intended audience. It seems to me that mathematical rigor is like clothing: in its style it ought to suit the occasion, and it diminishes comfort and restricts freedom of movement if it is either too loose or too tight. History and biography. There is an old Armenian saying, “He who lacks a sense of the past is condemned to live in the narrow darkness of his own generation.” Mathematics without history is mathematics stripped of its greatness: for, like the other arts—and mathematics is one of the supreme arts of civilization—it derives its grandeur from the fact of being a human creation. In an age increasingly dominated by mass culture and bureaucratic impersonality, I take great pleasure in knowing that the vital ideas of mathematics were not printed out by a computer or voted through by a committee, but instead were created by the solitary labor and individual genius of a few remarkable men. The many biographical notes in this book reflect my desire to convey something of the achievements and personal qualities of these astonishing human beings. Most of the longer notes are placed in the appendices, but each is linked directly to a specific contribution discussed in the text. These notes have as their subjects all but a few of the greatest mathematicians of the past three centuries: Fermat, Newton, the Bernoullis, Euler, Lagrange, Laplace, Fourier, Gauss, Abel, Poisson, Dirichlet, Hamilton, Liouville, Chebyshev, Hermite, Riemann, Minkowski, and Poincaré. As T. S. Eliot wrote in one of his essays, “Someone said: ‘The dead writers are remote from us because we know so much more than they did.’ Precisely, and they are that which we know.” History and biography are very complex, and I am painfully aware that scarcely anything in my notes is actually quite as simple as it may appear. I must also apologize for the many excessively brief allusions to xviii Preface to the First Edition mathematical ideas most student readers have not yet encountered. But with the aid of a good library, sufficiently interested students should be able to unravel most of them for themselves. At the very least, such efforts may help to impart a feeling for the immense diversity of classical mathematics—an aspect of the subject that is almost invisible in the average undergraduate curriculum. George F. Simmons Suggestions for the Instructor The following diagram gives the logical dependence of the chapters and suggests a variety of ways this book can be used, depending on the purposes of the course, the tastes of the instructor, and the backgrounds and needs of the students. 1. The nature of differential equations, separable equations 2. First-order equations 3. Second-order linear equations 9. Laplace transforms 4. Qualitative properties of solutions 5. Power series solutions and special functions 12. The calculus of variations 10. Systems of firstorder equations 11. Nonlinear equations 6. Fourier series and orthogonal functions 7. Partial differential equations and boundary value problems 8. Some special functions of mathematical physics 13. Existence and uniqueness theorems 14. Numerical methods xix xx Suggestions for the Instructor The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. Of course I do not here speak of that beauty that strikes the senses, the beauty of qualities and appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmonious order of the parts, and which a pure intelligence can grasp. —Henri Poincaré As a mathematical discipline travels far from its empirical source, or still more, if it is a second or third generation only indirectly inspired by ideas coming from “reality,“it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration. —John von Neumann Just as deduction should be supplemented by intuition, so the impulse to progressive generalization must be tempered and balanced by respect and love for colorful detail. The individual problem should not be degraded to the rank of special illustration of lofty general theories. In fact, general theories emerge from consideration of the specific, and they are meaningless if they do not serve to clarify and order the more particularized substance below. The interplay between generality and individuality, deduction and construction, logic and imagination—this is the profound essence of live mathematics. Any one or another of these aspects of mathematics can be at the center of a given achievement. In a far-reaching development all of them will be involved. Generally speaking, such a development will start from the “concrete” ground, then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy; after this flight comes the crucial test of landing and reaching specific goals in the newly surveyed low plains of individual “reality.” In brief, the flight into abstract generality must start from and return to the concrete and specific. —Richard Courant About the Author George Simmons has academic degrees from the California Institute of Technology, the University of Chicago, and Yale University. He taught at several colleges and universities before joining the faculty of Colorado College in 1962, where he is a Professor of Mathematics. He is also the author of Introduction to Topology and Modern Analysis (McGraw-Hill, 1963), Precalculus Mathematics in a Nutshell (Janson Publications, 1981), and Calculus with Analytic Geometry (McGraw-Hill, 1985). When not working or talking or eating or drinking or cooking, Professor Simmons is likely to be traveling (Western and Southern Europe, Turkey, Israel, Egypt, Russia, China, Southeast Asia), trout fishing (Rocky Mountain states), playing pocket billiards, or reading (literature, history, biography and autobiography, science, and enough thrillers to achieve enjoyment without guilt). xxi Chapter 1 The Nature of Differential Equations. Separable Equations 1 Introduction An equation involving one dependent variable and its derivatives with respect to one or more independent variables is called a differential equation. Many of the general laws of nature—in physics, chemistry, biology, and astronomy—find their most natural expression in the language of differential equations. Applications also abound in mathematics itself, especially in geometry, and in engineering, economics, and many other fields of applied science. It is easy to understand the reason behind this broad utility of differential equations. The reader will recall that if y = f(x) is a given function, then its derivative dy/dx can be interpreted as the rate of change of y with respect to x. In any natural process, the variables involved and their rates of change are connected with one another by means of the basic scientific principles that govern the process. When this connection is expressed in mathematical symbols, the result is often a differential equation. The following example may illuminate these remarks. According to Newton’s second law of motion, the acceleration a of a body of mass m is proportional to the total force F acting on it, with 1/m as the constant of proportionality, so that a = F/m or ma = F. (1) Suppose, for instance, that a body of mass m falls freely under the influence of gravity alone. In this case the only force acting on it is mg, where g is the acceleration due to gravity.1 If y is the distance down to the body from some fixed height, then its velocity v = dy/dt is the rate of change of position and its acceleration a = dv/dt = d2y/dt2 is the rate of change of velocity. With this notation, (1) becomes 1 g can be considered constant on the surface of the earth in most applications, and is approximately 32 feet per second per second (or 980 centimeters per second per second). 1 2 Differential Equations with Applications and Historical Notes m d2 y = mg dt 2 or d2 y = g. dt 2 (2) If we alter the situation by assuming that air exerts a resisting force proportional to the velocity, then the total force acting on the body is mg − k(dy/dt), and (1) becomes m d2 y dy = mg - k . 2 dt dt (3) Equations (2) and (3) are the differential equations that express the essential attributes of the physical processes under consideration. As further examples of differential equations, we list the following: dy = – ky; dt (4) d2 y = – ky; dt 2 (5) m (1 - x 2 ) x2 2 dy + 2xy = e – x ; dx (6) d2 y dy –5 + 6 y = 0; dx 2 dx (7) d2 y dy - 2x + p( p + 1)y = 0 ; 2 dx dx (8) d2 y dy +x + ( x 2 - p 2 )y = 0 . dx 2 dx (9) The dependent variable in each of these equations is y, and the independent variable is either t or x. The letters k, m, and p represent constants. An ordinary differential equation is one in which there is only one independent variable, so that all the derivatives occurring in it are ordinary derivatives. Each of these equations is ordinary. The order of a differential equation is the order of the 3 The Nature of Differential Equations highest derivative present. Equations (4) and (6) are first order equations, and the others are second order. Equations (8) and (9) are classical, and are called Legendre’s equation and Bessel’s equation, respectively. Each has a vast literature and a history reaching back hundreds of years. We shall study all of these equations in detail later. A partial differential equation is one involving more than one independent variable, so that the derivatives occurring in it are partial derivatives. For example, if w = f(x,y,z,t) is a function of time and the three rectangular coordinates of a point in space, then the following are partial differential equations of the second order: ¶ 2w ¶ 2w ¶ 2w + + = 0; ¶x 2 ¶y 2 ¶z 2 æ ¶ 2w ¶ 2w ¶ 2w ö ¶w ; a2 ç 2 + 2 + 2 ÷ = ¶y ¶z ø ¶t è ¶x æ ¶ 2w ¶ 2w ¶ 2w ö ¶ 2w a2 ç 2 + 2 + 2 ÷ = 2 . ¶y ¶z ø ¶t è ¶x These equations are also classical, and are called Laplace’s equation, the heat equation, and the wave equation, respectively. Each is profoundly significant in theoretical physics, and their study has stimulated the development of many important mathematical ideas. In general, partial differential equations arise in the physics of continuous media—in problems involving electric fields, fluid dynamics, diffusion, and wave motion. Their theory is very different from that of ordinary differential equations, and is much more difficult in almost every respect. For some time to come, we shall confine our attention exclusively to ordinary differential equations.2 2 The English biologist J. B. S. Haldane (1892–1964) has a good remark about the one-dimensional special case of the heat equation: “In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict new facts of the same kind. The catch in this criterion lies in the word ‘simplest.’ It is really an aesthetic canon such as we find implicit in our criticism of poetry or painting. The layman finds such a law as a2 ¶ 2 w ¶w = ¶x 2 ¶t much less simple than ‘it oozes,’ of which it is the mathematical statement. The physicist reverses this judgment, and his statement is certainly the more fruitful of the two, so far as prediction is concerned. It is, however, a statement about something very unfamiliar to the plain man, namely, the rate of change of a rate of change.” 4 Differential Equations with Applications and Historical Notes 2 General Remarks on Solutions The general ordinary differential equation of the nth order is æ dy d 2 y dn y ö F ç x, y , , 2 , … , n ÷ = 0, dx dx dx ø è (1) or, using the prime notation for derivatives, F(x, y, y′, y″,…,y(n)) = 0. Any adequate theoretical discussion of this equation would have to be based on a careful study of explicitly assumed properties of the function F. However, undue emphasis on the fine points of theory often tends to obscure what is really going on. We will therefore try to avoid being overly fussy about such matters—at least for the present. It is normally a simple task to verify that a given function y = y(x) is a solution of an equation like (1). All that is necessary is to compute the derivatives of y(x) and to show that y(x) and these derivatives, when substituted in the equation, reduce it to an identity in x. In this way we see that y = e2x and y = e3x are both solutions of the second order equation y″ − 5y′ + 6y = 0; (2) y = c1e2x + c2e3x (3) and, more generally, that is also a solution for every choice of the constants c1 and c2. Solutions of differential equations often arise in the form of functions defined implicitly, and sometimes it is difficult or impossible to express the dependent variable explicitly in terms of the independent variable. For instance, xy = log y + c (4) dy y2 = dx 1 - xy (5) is a solution of 5 The Nature of Differential Equations for every value of the constant c, as we can readily verify by differentiating (4) and rearranging the result.3 These examples also illustrate the fact that a solution of a differential equation usually contains one or more arbitrary constants, equal in number to the order of the equation. In most cases procedures of this kind are easy to apply to a suspected solution of a given differential equation. The problem of starting with a differential equation and finding a solution is naturally much more difficult. In due course we shall develop systematic methods for solving equations like (2) and (5). For the present, however, we limit ourselves to a few remarks on some of the general aspects of solutions. The simplest of all differential equations is dy = f ( x), dx (6) ò f (x) dx + c. (7) and we solve it by writing y= In some cases the indefinite integral in (7) can be worked out by the methods of calculus. In other cases it may be difficult or impossible to find a formula for this integral. It is known, for instance, that òe – x2 dx and ò sin x dx x cannot be expressed in terms of a finite number of elementary functions.4 If we recall, however, that ò f (x) dx is merely a symbol for a function (any function) with derivative f(x), then we can almost always give (7) a valid meaning by writing it in the form x y= ò f (t) dt + c. (8) x0 3 4 In calculus the notation In x is often used for the so-called natural logarithm, that is, the function loge x. In more advanced courses, however, this function is almost always denoted by the symbol log x. Any reader who is curious about the reasons for this should consult D. G. Mead, “Integration,” Am. Math. Monthly, vol. 68, pp. 152–156 (1961). For additional details, see G. H. Hardy, The Integration of Functions of a Single Variable, Cambridge University Press, London, 1916; or J. F. Ritt, Integration in Finite Terms, Columbia University Press, New York, 1948. 6 Differential Equations with Applications and Historical Notes The crux of the matter is that this definite integral is a function of the upper limit x (the t under the integral sign is only a dummy variable) which always exists when the integrand is continuous over the range of integration, and that its derivative is f(x).5 The so-called separable equations, or equations with separable variables, are at the same level of simplicity as (6). These are differential equations that can be written in the form dy = f ( x) g( y ), dx where the right side is a product of two functions each of which depends on only one of the variables. In such a case we can separate the variables by writing dy = f ( x) dx, g( y ) and then solve the original equation by integrating: dy ò g(y) = ò f (x) dx + c. These are simple differential equations to deal with in the sense that the problem of solving them can be reduced to the problem of integration, even though the indicated integrations can be difficult or impossible to carry out explicitly. The general first order equation is the special case of (1) which corresponds to taking n = 1: dy ö æ F ç x , y , ÷ = 0. dx ø è (9) We normally expect that an equation like this will have a solution, and that this solution—like (7) and (8)—will contain one arbitrary constant. However, 2 æ dy ö ç dx ÷ + 1 = 0 è ø 5 This statement is one form of the fundamental theorem of calculus. 7 The Nature of Differential Equations has no real-valued solutions at all, and 2 æ dy ö 2 ç dx ÷ + y = 0 è ø has only the single solution y = 0 (which contains no arbitrary constants). Situations of this kind raise difficult theoretical questions about the existence and nature of solutions of differential equations. We cannot enter here into a full discussion of these questions, but it may clarify matters if we give an intuitive description of a few of the basic facts. For the sake of simplicity, let us assume that (9) can be solved for dy/dx: dy = f ( x , y ). dx (10) We also assume that f(x,y) is a continuous function throughout some rectangle R in the xy plane. The geometric meaning of a solution of (10) can best be understood as follows (Figure 1). If P0 = (x0,y0) is a point in R, then the number æ dy ö ç ÷ = f ( x0 , y 0 ) è dx øP0 y R P2 P0 P1 x FIGURE 1 8 Differential Equations with Applications and Historical Notes determines a direction at P0. Now let P1 = (x1 y1) be a point near P0 in this direction, and use æ dy ö ç ÷ = f ( x1 , y1 ) è dx øP1 to determine a new direction at P1. Next, let P2 = (x2, y2) be a point near P1 in this new direction, and use the number æ dy ö ç ÷ = f ( x2 , y 2 ) è dx øP2 to determine yet another direction at P2. If we continue this process, we obtain a broken line with points scattered along it like beads; and if we now imagine that these successive points move closer to one another and become more numerous, then the broken line approaches a smooth curve through the initial point P0. This curve is a solution y = y(x) of equation (10); for at each point (x,y) on it, the slope is given by f(x,y)—and this is precisely the condition required by the differential equation. If we start with a different initial point, then in general we obtain a different curve (or solution). Thus the solutions of (10) form a family of curves, called integral curves.6 Furthermore, it appears to be a reasonable guess that through each point in R there passes just one integral curve of (10). This discussion is intended only to lend plausibility to the following precise statement. Theorem A. (Picard’s theorem.) If f(x,y) and ∂f/∂y are continuous functions on a closed rectangle R, then through each point (x0, y0) in the interior of R there passes a unique integral curve of the equation dy/dx = f(x,y). If we consider a fixed value of x0 in this theorem, then the integral curve that passes through (x0, y0) is fully determined by the choice of y0. In this way we see that the integral curves of (10) constitute what is called a oneparameter family of curves. The equation of this family can be written in the form y = y(x, c), (11) where different choices of the parameter c yield different curves in the family. The integral curve that passes through (x0, y0) corresponds to the value of 6 Solutions of a differential equation are sometimes called integrals of the equation because the problem of finding them is more or less an extension of the ordinary problem of integration. 9 The Nature of Differential Equations c for which y0 = y(x0,c). If we denote this number by c0, then (11) is called the general solution of (10), and y = y(x, c0) is called the particular solution that satisfies the initial condition y = y0 when x = x0. The essential feature of the general solution (11) is that the constant c in it can be chosen so that an integral curve passes through any given point of the rectangle under consideration. Picard’s theorem is proved in Chapter 13. This proof is quite complicated, and is probably best postponed until the reader has had considerable experience with the more straightforward parts of the subject. The theorem itself can be strengthened in various directions by weakening its hypotheses; it can also be generalized to refer to nth order equations solvable for the nth order derivative. Detailed descriptions of these results would be out of place in the present context, and we content ourselves for the time being with this informal discussion of the main ideas. In the rest of this chapter we explore some of the ways in which differential equations arise in scientific applications. Problems 1. Verify that the following functions (explicit or implicit) are solutions of the corresponding differential equations: (a) y = x2 + c y′ = 2x; 2 (b) y = cx xy′ = 2y; (c) y2 = e2x + c yy′ = e2x; (d) y = cekx y′ = ky; (e) y = c1 sin 2x + c2 cos 2x y″ + 4y = 0; 2x −2x (f) y = c1.e + c2e y″ − 4y = 0; (g) y = c1 sinh 2x + c2 cosh 2x y″ − 4y = 0; (h) y = sin−1 xy (i) y = x tan x (j) x2 = 2y2 log y; (k) y2 = x2 − cx (1) y = c2 + c/x xy¢ + y = y¢ 1 – x 2 y 2 ; xy′ = y + x2 + y2; xy y¢ = 2 ; x + y2 2xyy′ = x2 + y2; y + xy′ = x4(y′)2; 10 Differential Equations with Applications and Historical Notes (m) y = cey/x y′ = y2/(xy − x2); (n) y + sin y = x (y cos y − sin y + x)y′ = y; −1 (o) x + y = tan y 1 + y2 + y2y′ = 0. 2. Find the general solution of each of the following differential equations: (a) y′ = e3x − x; (b) xy′ = 1; (c) y′ = xex2; (d) y′ = sin−1 x; (e) (1 + x)y′ = x; (f) (1 + x2)y′ = x; (g) (1 + x3)y′ = x; (h) (1 + x2)y′ = tan−1x; (i) xyy′ = y − 1; (j) x5y′ + y5 = 0; (k) xy′ = (1 − 2x2) tan y; (1) y′ = 2xy; (m) y′ sin y = x2; (n) y′ sin x = 1; (o) y′ + y tan x = 0; (p) y′ − y tan x = 0; (q) (1 + x2) dy + (1 + y2) dx = 0; (r) y log y dx − x dy = 0. 3. For each of the following differential equations, find the particular solution that satisfies the given initial condition: (a) y′ = xex, y = 3 when x = 1; (b) y′ = 2 sin x cos x, y = 1 when x = 0; (c) y′ = log x, y = 0 when x = e; (d) (x2 − l)y′ = 1, y = 0 when x = 2; (e) x(x2 − 4)y′ = 1, y = 0 when x = 1; (f) (x + 1)(x2 + l)y′ = 2x2 + x, y = 1 when x = 0. 4. For each of the following differential equations, find the integral curve that passes through the given point: (a) y′ = e3x−2y, (0, 0); (b) x dy = (2x2 + 1) dx, (1, 1); (c) e–y dx + (1 + x2) dy = 0, (0, 0); (d) 3 cos 3x cos 2y dx − 2 sin 3x sin 2y dy = 0, (π/12,π/8); The Nature of Differential Equations 11 (e) y′ = ex cos x, (0,0); (f) xyy′ = (x + l)(y + 1), (1,0). 5. Show that y = e x 2 x òe 0 – t2 dt is a solution of y′ = 2xy + 1. 6. For the differential equation (2), namely, y″ − 5y′ + 6y = 0, carry out the detailed calculations needed to verify the assertions in the text that (a) y = e2x and y = e3x are both solutions; and (b) y = c1e2x + c2 e3x is a solution for every choice of the constants c1 and c2. Remark: In studying a book like this, a student should never slide past assertions of this kind—involving such phrases as “we see” or “as we can readily verify”—without personally checking their validity. The mere fact that something is in print does not mean it is necessarily true. Cultivate skepticism as a healthy state of mind, as you would physical fitness; accept nothing on the authority of this writer or any other until you have understood it fully for yourself. 7. In the spirit of Problem 6, verify that (4) is a solution of the differential equation (5) for every value of the constant c. 8. For what values of the constant m will y = emx be a solution of the differential equation 2 y¢¢¢ + y¢¢ - 5 y¢ + 2 y = 0 ? Use the ideas in Problem 6 to find a solution containing three arbitrary constants c1, c2, c3. 3 Families of Curves. Orthogonal Trajectories We have seen that the general solution of a first order differential equation normally contains one arbitrary constant, called a parameter. When this parameter is assigned various values, we obtain a one-parameter family of curves. Each of these curves is a particular solution, or integral curve, of the given differential equation, and all of them together constitute its general solution. Conversely, as we might expect, the curves of any one-parameter family are integral curves of some first order differential equation. If the family is f(x, y, c) = 0, (l) 12 Differential Equations with Applications and Historical Notes then its differential equation can be found by the following steps. First, differentiate (1) implicitly with respect to x to get a relation of the form dy ö æ g ç x, y , , c = 0. dx ÷ø è (2) Next, eliminate the parameter c from (1) and (2) to obtain dy ö æ F ç x, y , ÷ = 0 dx ø è (3) as the desired differential equation. For example, x 2 + y2 = c 2 (4) is the equation of the family of all circles with centers at the origin (Figure 2). On differentiation with respect to x this becomes 2x + 2 y dy = 0; dx y x FIGURE 2 13 The Nature of Differential Equations and since c is already absent, there is no need to eliminate it and x+y dy =0 dx (5) is the differential equation of the given family of circles. Similarly, x2 + y2 = 2cx (6) is the equation of the family of all circles tangent to the y-axis at the origin (Figure 3). When we differentiate this with respect to x, we obtain dy = 2c dx 2x + 2 y or x+y dy =c dx (7) y x FIGURE 3 14 Differential Equations with Applications and Historical Notes The parameter c is still present, so it is necessary to eliminate it by combining (6) and (7). This yields dy y 2 – x 2 = 2xy dx (8) as the differential equation of the family (6). As an interesting application of these procedures, we consider the problem of finding orthogonal trajectories. To explain what this problem is, we observe that the family of circles represented by (4) and the family y = mx of straight lines through the origin (the dotted lines in Figure 2) have the following property: each curve in either family is orthogonal (i.e., perpendicular) to every curve in the other family. Whenever two families of curves are related in this way, each is said to be a family of orthogonal trajectories of the other. Orthogonal trajectories are of interest in the geometry of plane curves, and also in certain parts of applied mathematics. For instance, if an electric current is flowing in a plane sheet of conducting material, then the lines of equal potential are the orthogonal trajectories of the lines of current flow. In the example of the circles centered on the origin, it is geometrically obvious that the orthogonal trajectories are the straight lines through the origin, and conversely. In order to cope with more complicated situations, however, we need an analytic method for finding orthogonal trajectories. Suppose that dy = f (x, y) dx (9) is the differential equation of the family of solid curves in Figure 4. These curves are characterized by the fact that at any point (x,y) on any one of them the slope is given by f(x,y). The dotted orthogonal trajectory through the same point, being orthogonal to the first curve, has as its slope the negative reciprocal of the first slope. Thus, along any orthogonal trajectory, we have dy/dx = −1/f(x,y) or - dx = f ( x , y ). dy (10) Our method of finding the orthogonal trajectories of a given family of curves is therefore as follows: first, find the differential equation of the family; next, replace dy/dx by –dx/dy to obtain the differential equation of the orthogonal trajectories; and finally, solve this new differential equation. 15 The Nature of Differential Equations Slope = –1/f (x, y) Slope = f (x, y) (x, y) FIGURE 4 If we apply this method to the family of circles (4) with differential equation (5), we get æ dx ö x + yç– ÷=0 è dy ø or dy y = dx x (11) as the differential equation of the orthogonal trajectories. We can now separate the variables in (11) to obtain dy dx = , y x which on direct integration yields log y = log x + log c or y = cx as the equation of the orthogonal trajectories. 16 Differential Equations with Applications and Historical Notes y dr rdθ ψ dθ r θ x FIGURE 5 It is often convenient to express the given family of curves in terms of polar coordinates. In this case we use the fact that if ψ is the angle from the polar radius to the tangent, then tan ψ = r dθ/dr (Figure 5). By the above discussion, we replace this expression in the differential equation of the given family by its negative reciprocal, –dr/r dθ, to obtain the differential equation of the orthogonal trajectories. As an illustration of the value of this technique, we find the orthogonal trajectories of the family of circles (6). If we use rectangular coordinates, it follows from (8) that the differential equation of the orthogonal trajectories is dy 2xy . = 2 dx x – y 2 (12) Unfortunately, the variables in (12) cannot be separated, so without additional techniques for solving differential equations we can go no further in this direction. However, if we use polar coordinates, the equation of the family (6) can be written as r = 2c cos θ. (13) The Nature of Differential Equations 17 From this we find that dr = –2c sin q, dq (14) and after eliminating c from (13) and (14) we arrive at cos q rdq =– dr sin q as the differential equation of the given family. Accordingly, rdq sin q = dr cos q is the differential equation of the orthogonal trajectories. In this case the variables can be separated, yielding dr cos q dq = ; r sin q and after integration this becomes log r = log (sin θ) + log 2c, so that r = 2c sin θ (15) is the equation of the orthogonal trajectories. It will be noted that (15) is the equation of the family of all circles tangent to the x-axis at the origin (see the dotted curves in Figure 3). In Chapter 2 we develop a number of more elaborate procedures for solving first order equations. Since our present attention is directed more at applications than formal techniques, all the problems given in this chapter are solvable by the method of separation of variables illustrated above. Problems 1. Sketch each of the following families of curves, find the orthogonal trajectories, and add them to the sketch: (a) xy = c; (b) y = cx2; 18 Differential Equations with Applications and Historical Notes (c) r = c (1 + cos θ); (d) y = cex. 2. What are the orthogonal trajectories of the family of curves (a) y = cx4; (b) y = cxn where n is any positive integer? In each case, sketch both families of curves. What is the effect on the orthogonal trajectories of increasing the exponent n? 3. Show that the method for finding orthogonal trajectories in polar coordinates can be expressed as follows. If dr/dθ = F(r, θ) is the differential equation of the given family of curves, then dr/dθ = –r2/F(r, θ) is the differential equation of the orthogonal trajectories. Apply this method to the family of circles r = 2c sin θ. 4. Use polar coordinates to find the orthogonal trajectories of the family of parabolas r = c/(1 − cos θ), c > 0. Sketch both families of curves. 5. Sketch the family y2 = 4c(x + c) of all parabolas with axis the x-axis and focus at the origin, and find the differential equation of the family. Show that this differential equation is unaltered when dy/dx is replaced by –dx/dy. What conclusion can be drawn from this fact? 6. Find the curves that satisfy each of the following geometric conditions: (a) The part of the tangent cut off by the axes is bisected by the point of tangency. (b) The projection on the x-axis of the part of the normal between (x,y) and the x-axis has length 1. (c) The projection on the x-axis of the part of the tangent between (x,y) and the x-axis has length 1. (d) The part of the tangent between (x, y) and the x-axis is bisected by the y-axis. (e) The part of the normal between (x, y) and the y-axis is bisected by the x-axis. (f) (x, y) is equidistant from the origin and the point of intersection of the normal with the x-axis. (g) The polar angle θ equals the angle ψ from the polar radius to the tangent. (h) The angle ψ from the polar radius to the tangent is constant. 7. A curve rises from the origin in the xy-plane into the first quadrant. The area under the curve from (0, 0) to (x, y) is one-third the area of the rectangle with these points as opposite vertices. Find the equation of the curve. 8. Three vertices of a rectangle of area A lie on the x-axis, at the origin, and on the y-axis. If the fourth vertex moves along a curve y = y(x) in the first quadrant in such a way that the rate of change of A with respect to x is proportional to A, find the equation of the curve. 19 The Nature of Differential Equations 9. A saddle without a saddle-horn (pommel) has the shape of the surface z = y2 − x2. It is lying outdoors in a rainstorm. Find the paths along which raindrops will run down the saddle. Draw a sketch and use it to convince yourself that your answer is reasonable. 10. Find the differential equation of each of the following one-parameter families of curves: (a) y = x sin (x + c); (b) all circles through (1, 0) and (−1, 0); (c) all circles with centers on the line y = x and tangent to both axes; (d) all lines tangent to the parabola x2 = 4y (hint: the slope of the tangent line at (2a, a2) is a); (e) all lines tangent to the unit circle x2 + y2 = 1. 11. In part (d) of Problem 10, show that the parabola itself is an integral curve of the differential equation of the family of all its tangent lines, and that therefore through each point of this parabola there pass two integral curves of this differential equation. Do the same for the unit circle in part (e) of Problem 10. 4 Growth, Decay, Chemical Reactions, and Mixing We remind the student that the number e is often defined by the limit n 1ö æ e = lim ç 1 + ÷ , n ®¥ è nø or slightly more generally (put h = 1/n), by the limit e = lim (1 + h)1 h. h ®0 (1) In words, this says that e is the limit of 1 plus a small number, raised to the power of the reciprocal of the small number, as that small number approaches 0. We recall from calculus that the importance of the number e lies mainly in the fact that the exponential function y = ex is unchanged by differentiation: d x e = e x. dx 20 Differential Equations with Applications and Historical Notes An equivalent statement is that y = ex is a solution of the differential equation dy = y. dx More generally, if k is any given nonzero constant, then all of the functions y = cekx are solutions of the differential equation dy = ky. dx (2) This is easy to verify by differentiation, and can also be discovered by separating the variables and integrating: dy = k dx , y log y = kx + c0 , y = e kx + c0 = e c0 e kx = ce kx . Further, it is not difficult to show that these functions are the only solutions of equation (2) [see Problem 1]. In this section we discuss a surprisingly wide variety of applications of these facts to a number of different sciences. Example 1. Continuously compounded interest. If P dollars is deposited in a bank that pays an interest rate of 6 percent per year, compounded semiannually, then after t years the accumulated amount is A = P (1 + 0.03)2t. More generally, if the interest rate is 100k percent (k = 0.06 for 6 percent), and if this interest is compounded n times a year, then after t years the accumulated amount is nt kö æ A = Pç1 + ÷ . nø è If n is now increased indefinitely, so that the interest is compounded more and more frequently, then we approach the limiting case of continuously compounded interest.7 To find the formula for A under these circumstances, we observe that (1) yields 7 Many banks pay interest daily, which corresponds to n = 365. This number is large enough to make continuously compounded interest a very accurate model for what actually happens. 21 The Nature of Differential Equations kt nt nk éæ kö kö ù æ kt ç 1 + ÷ = êç 1 + ÷ ú ® e , nø n ø úû è ëêè so A = Pekt. (3) We describe this situation by saying that the amount A grows exponentially, or provides an example of exponential growth. To understand the meaning of the constant k from a different point of view, we differentiate (3) to obtain dA = Pke kt = kA . dt If we write this differential equation for A in the form dA A = k, dt then we see that k can be thought of as the fractional change in A per unit time, and 100k is the percentage change in A per unit time. Example 2. Population growth. Suppose that x0 bacteria are placed in a nutrient solution at time t = 0, and that x = x(t) is the population of the colony at a later time t. If food and living space are unlimited, and if as a consequence the population at any moment is increasing at a rate proportional to the population at that moment, find x as a function of t.8 Since the rate of increase of x is proportional to x itself, we can write down the differential equation dx = kx . dt By separating the variables and integrating, we get dx = k dt , x log x = kt + c. Since x = x0 when t = 0, we have c = logx0, so log x = kt + log x0 and x = x0 ekt. (4) We therefore have another example of exponential growth. 8 Briefly, this assumption about the rate means that we expect twice as many “births” in a given short interval of time when twice as many bacteria are present. 22 Differential Equations with Applications and Historical Notes To make these ideas more concrete, let us assume for the sake of discussion that the total human population of the earth grows in this way. According to the United Nations demographic experts, this population is increasing at an overall rate of approximately 2 percent per year, so k = 0.02 = 1/50 and (4) becomes x = x0 et/50. (5) To find the “doubling time” T, that is, the time needed for the total number of people in the world to increase by a factor of 2, we replace (5) by 2x0 = x0 eT/50. This yields T/50 = log 2, so T = 50 log 2 ≅ 34.65 years, since log 2 ≅ 0.693.9 Example 3. Radioactive decay. If molecules of a certain kind have a tendency to decompose into smaller molecules at a rate unaffected by the presence of other substances, then it is natural to expect that the number of molecules of this kind that will decompose in a unit of time will be proportional to the total number present. A chemical reaction of this type is called a first order reaction. Suppose, for instance, that x0 grams of matter are present initially, and decompose in a first order reaction. If x is the number of grams present at a later time t, then the principle stated above yields the following differential equation: – dx = kx , dt k > 0. (6) [Since dx/dt is the rate of growth of x, –dx/dt is its rate of decay, and (6) says that the rate of decay is proportional to x.] If we separate the variables in (6) and integrate, we obtain dx = – k dt , x 9 log x = – kt + c . It is worth mentioning that the population of the industrialized nations is increasing at a rate somewhat less than 2 percent, while that of the third world nations is increasing at a rate greater than 2 percent. From the point of view of the development of the human race and its social and political institutions over the next several centuries, this is perhaps the most important single fact about our contemporary world. 23 The Nature of Differential Equations The initial condition x = x0 when t = 0 (7) gives c = log x0, so log x = –kt + log x0 and x = x0 e –kt. (8) This function is therefore the solution of the differential equation (6) that satisfies the initial condition (7). Its graph is given in Figure 6. The positive constant k is called the rate constant, for its value is clearly a measure of the rate at which the reaction proceeds. As we know from Example 1, k can be thought of as the fractional loss of x per unit time. Very few first order chemical reactions are known, and by far the most important of these is radioactive decay. It is convenient to express the rate of decay of a radioactive element in terms of its half-life, which is the time required for a given quantity of the element to diminish by a factor of one-half. If we replace x by x0/2 in formula (8), then we get the equation x0 = x0 e – kT 2 for the half-life T, so kT = log 2. If either k or T is known from observation or experiment, this equation enables us to find the other. The situation discussed here is an example of exponential decay. This phrase refers only to the form of the function (8) and the manner in which the quantity x diminishes, and not necessarily to the idea that something or other is disintegrating. x x0 ½ x0 T FIGURE 6 t 24 Differential Equations with Applications and Historical Notes Example 4. Mixing. A tank contains 50 gallons of brine in which 75 pounds of salt are dissolved. Beginning at time t = 0, brine containing 3 pounds of salt per gallon flows in at the rate of 2 gallons per minute, and the mixture (which is kept uniform by stirring) flows out at the same rate. When will there be 125 pounds of dissolved salt in the tank? How much dissolved salt is in the tank after a long time? If x = x(t) is the number of pounds of dissolved salt in the tank at time t ≥ 0, then the concentration at that time is x/50 pounds per gallon. The rate of change of x is dx = rate at which salt enters tank − rate at which salt leaves tank. dt Since rate of entering = 3 · 2 = 6 lb/min and rateof leaving = ( x 50) × 2 = x 1b min , 25 we have dx x 150 – x =6– = . dt 25 25 Separating variables and integrating give dx 1 dt = 150 – x 25 and log(150 – x) = – 1 t + c. 25 Since x = 75 when t = 0, we see that c = log 75, so log(150 - x) = - 1 t + log 75 , 25 and therefore 150 − x = 75e–t/25 or x = 75(2 − e–/t25). This tells us that x = 125 implies et/25 = 3 or t/25 = log 3. We conclude that x = 125 pounds after t = 25 log 3 ≅ 27.47 minutes, since log 3 ≅ 1.0986. Also, when t is large we see that x is nearly 75 ? 2 = 150 pounds, as common sense tells us without calculation. The Nature of Differential Equations 25 The ideas discussed in Example 3 are the basis for a scientific tool of fairly recent development which has been of great significance for geology and archaeology. In essence, radioactive elements occurring in nature (with known half-lives) can be used to assign dates to events that took place from a few thousand to a few billion years ago. For example, the common isotope of uranium decays through several stages into helium and an isotope of lead, with a halflife of 4.5 billion years. When rock containing uranium is in a molten state, as in lava flowing from the mouth of a volcano, the lead created by this decay process is dispersed by currents in the lava; but after the rock solidifies, the lead is locked in place and steadily accumulates alongside the parent uranium. A piece of granite can be analyzed to determine the ratio of lead to uranium, and this ratio permits an estimate of the time that has elapsed since the critical moment when the granite crystallized. Several methods of age determination involving the decay of thorium and the isotopes of uranium into the various isotopes of lead are in current use. Another method depends on the decay of potassium into argon, with a half-life of 1.3 billion years; and yet another, preferred for dating the oldest rocks, is based on the decay of rubidium into strontium, with a half-life of 50 billion years. These studies are complex and susceptible to errors of many kinds; but they can often be checked against one another, and are capable of yielding reliable dates for many events in geological history linked to the formation of igneous rocks. Rocks tens of millions of years old are quite young, ages ranging into hundreds of millions of years are common, and the oldest rocks yet discovered are upwards of 3 billion years old. This of course is a lower limit for the age of the earth’s crust, and so for the age of the earth itself. Other investigations, using various types of astronomical data, age determinations for minerals in meteorites, and so on, have suggested a probable age for the earth of about 4.5 billion years.10 The radioactive elements mentioned above decay so slowly that the methods of age determination based on them are not suitable for dating events that took place relatively recently. This gap was filled by Willard Libby’s discovery in the late 1940s of radiocarbon, a radioactive isotope of carbon with a half-life of about 5600 years. By 1950 Libby and his associates had developed the technique of radiocarbon dating, which added a second hand to the slowmoving geological clocks described above and made it possible to date events in the later stages of the Ice Age and some of the movements and activities of prehistoric man. The contributions of this technique to late Pleistocene geology and archaeology have been spectacular. In brief outline, the facts and principles involved are these. Radiocarbon is produced in the upper atmosphere by the action of cosmic ray neutrons on nitrogen. This radiocarbon is oxidized to carbon dioxide, which in turn is mixed by the winds with the nonradioactive carbon dioxide already present. Since radiocarbon is constantly being formed and constantly decomposing 10 For a full discussion of these matters, as well as many other methods and results of the science of geochronology, see F. E. Zeuner, Dating the Past, 4th ed., Methuen, London, 1958. 26 Differential Equations with Applications and Historical Notes back into nitrogen, its proportion to ordinary carbon in the atmosphere has long since reached an equilibrium state. All air-breathing plants incorporate this proportion of radiocarbon into their tissues, as do the animals that eat these plants. This proportion remains constant as long as a plant or animal lives; but when it dies it ceases to absorb new radiocarbon, while the supply it has at the time of death continues the steady process of decay. Thus, if a piece of old wood has half the radioactivity of a living tree, it lived about 5600 years ago, and if it has only a fourth this radioactivity, it lived about 11,200 years ago. This principle provides a method for dating any ancient object of organic origin, for instance, wood, charcoal, vegetable fiber, flesh, skin, bone, or horn. The reliability of the method has been verified by applying it to the heartwood of giant sequoia trees whose growth rings record 3000 to 4000 years of life, and to furniture from Egyptian tombs whose age is also known independently. There are technical difficulties, but the method is now felt to be capable of reasonable accuracy as long as the periods of time involved are not too great (up to about 50,000 years). Radiocarbon dating has been applied to thousands of samples, and laboratories for carrying on this work number in the dozens. Among the more interesting age estimates are these: linen wrappings from the Dead Sea scrolls of the Book of Isaiah, recently found in a cave in Palestine and thought to be first or second century b.c., 1917 ± 200 years; charcoal from the Lascaux cave in southern France, site of the remarkable prehistoric paintings, 15,516 ± 900 years; charcoal from the prehistoric monument at Stonehenge, in southern England, 3798 ± 275 years; charcoal from a tree burned at the time of the volcanic explosion that formed Crater Lake in Oregon, 6453 ± 250 years. Campsites of ancient man throughout the Western Hemisphere have been dated by using pieces of charcoal, fiber sandals, fragments of burned bison bone, and the like. The results suggest that human beings did not arrive in the New World until about the period of the last Ice Age, roughly 25,000 years ago, when the level of the water in the oceans was substantially lower than it now is and they could have walked across the Bering Straits from Siberia to Alaska.11 Problems 1. If k is a given nonzero constant, show that the functions y = cekx are the only solutions of the differential equation dy = ky. dx 11 Libby won the 1960 Nobel Prize for chemistry as a consequence of the work described above. His own account of the method, with its pitfalls and conclusions, can be found in his book Radiocarbon Dating, 2d ed., University of Chicago Press, 1955. The Nature of Differential Equations 27 Hint: Assume that f(x) is a solution of this equation and show that f(x)/ekx is a constant. 2. Suppose that P dollars is deposited in a bank that pays interest at an annual rate of r percent compounded continuously. (a) Find the time T required for this investment to double in value as a function of the interest rate r. (b) Find the interest rate that must be obtained if the investment is to double in value in 10 years. 3. A bright young executive with foresight but no initial capital makes constant investments of D dollars per year at an annual interest rate of 100k percent. Assume that the investments are made continuously and that interest is compounded continuously. (a) Find the accumulated amount A at any time t. (b) If the interest rate is 6 percent, what must D be if 1 million dollars is to be available for retirement 40 years later? (c) If the bright young executive is bright enough to find a safe investment opportunity paying 10 percent, what must D be to achieve the same result of 1 million dollars 40 years later? (It is worth noticing that if this amount of money is simply squirreled away without interest each year for 40 years, the grand total will be less than $80,000.) 4. A newly retired person invests total life savings of P dollars at an interest rate of 100k percent per year, compounded continuously. Withdrawals for living expenses are made continuously at a rate of W dollars per year. (a) Find the accumulated amount A at any time t. (b) Find the withdrawal rate W0 at which A will remain constant. (c) If W is greater than the value W0 found in part (b), then A will decrease and ultimately disappear. How long will this take? (d) Find the time in part (c) if the interest rate is 5 percent and W = 2W0. 5. A certain stock market tycoon has a fortune that increases at a rate proportional to the square of its size at any time. If he had 10 million dollars a year ago, and has 20 million dollars today, how wealthy will he be in 6 months? In a year? 6. A bacterial culture of population x is known to have a growth rate proportional to x itself. Between 6 p.m. and 7 p.m. the population triples. At what time will the population become 100 times what it was at 6 p.m.? 7. The population of a certain mining town is known to increase at a rate proportional to itself. After 2 years the population doubled, and after 1 more year the population was 10,000. What was the original population? 28 Differential Equations with Applications and Historical Notes 8. It is estimated by experts on agriculture that one-third of an acre of land is needed to provide food for one person on a continuing basis. It is also estimated that there are 10 billion acres of arable land on earth, and that therefore a maximum population of 30 billion people can be sustained if no other sources of food are known. The total world population at the beginning of 1970 was 3.6 billion. Assuming that the population continues to increase at the rate of 2 percent per year, when will the earth be full? What will be the population in the year 2000? 9. A mold grows at a rate proportional to the amount present. At the beginning the amount was 2 grams. In 2 days the amount has increased to 3 grams. (a) If x = x(t) is the amount of the mold at time t, show that x = 2(3/2)t/2. (b) Find the amount at the end of 10 days. 10. In Example 2, assume that living space for the colony of bacteria is limited and food is supplied at a constant rate, so that competition for food and space acts in such a way that ultimately the population will stabilize at a constant level x1 (x1 can be thought of as the largest population sustainable by this environment). Assume further that under these conditions the population grows at a rate proportional to the product of x and the difference x1 − x, and find x as a function of t. Sketch the graph of this function. When is the population increasing most rapidly? 11. Nuclear fission produces neutrons in an atomic pile at a rate proportional to the number of neutrons present at any moment. If n0 neutrons are present initially, and n1 and n2 neutrons are present at times t1 and t2, show that t2 t1 æ n1 ö æ n2 ö çn ÷ =çn ÷ . è 0ø è 0ø 12. If half of a given quantity of radium decomposes in 1600 years, what percentage of the original amount will be left at the end of 2400 years? At the end of 8000 years? 13. If the half-life of a radioactive substance is 20 days, how long will it take for 99 percent of the substance to decay? 14. A field of wheat teeming with grasshoppers is dusted with an insecticide having a kill rate of 200 per 100 per hour. What percentage of the grasshoppers are still alive 1 hour later? 15. Uranium-238 decays at a rate proportional to the amount present. If x1 and x2 grams are present at times t1 and t2, show that the half-life is (t2 - t1 )log 2 . log( x1/x2 ) The Nature of Differential Equations 29 16. Suppose that two chemical substances in solution react together to form a compound. If the reaction occurs by means of the collision and interaction of the molecules of the substances, then we expect the rate of formation of the compound to be proportional to the number of collisions per unit time, which in turn is jointly proportional to the amounts of the substances that are untransformed. A chemical reaction that proceeds in this manner is called a second order reaction, and this law of reaction is often referred to as the law of mass action. Consider a second order reaction in which x grams of the compound contain ax grams of the first substance and bx grams of the second, where a + b = 1. If there are aA grams of the first substance present initially, and bB grams of the second, and if x = 0 when t = 0, find x as a function of the time t.12 17. Many chemicals dissolve in water at a rate which is jointly proportional to the amount undissolved and to the difference between the concentration of a saturated solution and the concentration of the actual solution. For a chemical of this kind placed in a tank containing G gallons of water, find the amount x undissolved at time t if x = x0 when t = 0 and x = when t = t1, and if S is the amount dissolved in the tank when the solution is saturated. 18. Suppose that a given population can be divided into two groups: those who have a certain infectious disease, and those who do not have it but can catch it by having contact with an infected person. If x and y are the proportions of infected and uninfected people, then x + y = 1. Assume that (1) the disease spreads by the contacts just mentioned between sick people and well people, (2) that the rate of spread dx/dt is proportional to the number of such contacts, and (3) that the two groups mingle freely with each other, so that the number of contacts is jointly proportional to x and y. If x = x0 when t = 0, find x as a function of t, sketch the graph, and use this function to show that ultimately the disease will spread through the entire population. 19. A tank contains 100 gallons of brine in which 40 pounds of salt are dissolved. It is desired to reduce the concentration of salt to 0.1 pounds per gallon by pouring in pure water at the rate of 5 gallons per minute and allowing the mixture (which is kept uniform by stirring) to flow out at the same rate. How long will this take? 20. An aquarium contains 10 gallons of polluted water. A filter is attached to this aquarium which drains off the polluted water at the rate of 5 gallons per hour and replaces it at the same rate by pure water. How long does it take to reduce the pollution to half its initial level? 12 Students who are especially interested in first and second order chemical reactions will find a much more detailed discussion by Linus Pauling, probably the greatest chemist of the twentieth century, in his book General Chemistry, 3d ed., W. H. Freeman and Co., San Francisco, 1970. See particularly the chapter “The Rate of Chemical Reactions,” which is Chapter 16 in the 3d edition. 30 Differential Equations with Applications and Historical Notes 21. A party is being held in a room that contains 1800 cubic feet of air which is originally free of carbon monoxide. Beginning at time t = 0 several people start smoking cigarettes. Smoke containing 6 percent carbon monoxide is introduced into the room at the rate of 0.15 cubic feet/min, and the well-circulated mixture leaves at the same rate through a small open window. Extended exposure to a carbon monoxide concentration as low as 0.00018 can be dangerous. When should a prudent person leave this party? 22. According to Lambert’s law of absorption, the percentage of incident light absorbed by a thin layer of translucent material is proportional to the thickness of the layer.13 If sunlight falling vertically on ocean water is reduced to one-half its initial intensity at a depth of 10 feet, at what depth is it reduced to one-sixteenth its initial intensity? Solve this problem by merely thinking about it, and also by setting up and solving a suitable differential equation. 23. If sunlight falling vertically on lake water is reduced to three-fifths its initial intensity I0 at a depth of 15 feet, find its intensity at depths of 30 feet and 60 feet. Find the intensity at a depth of 50 feet. 24. Consider a column of air of cross-sectional area 1 square inch extending from sea level up to “infinity.” The atmospheric pressure p at an altitude h above sea level is the weight of the air in this column above the altitude h. Assuming that the density of the air is proportional to the pressure, show that p satisfies the differential equation dp = -cp, dh c > 0, and obtain the formula p = p0 e−ch, where p0 is the atmospheric pressure at sea level. 25. Assume that the rate at which a hot body cools is proportional to the difference in temperature between it and its surroundings (Newton’s law of cooling14). A body is heated to 110°C and placed in air at 10°C. After 1 hour its temperature is 60°C. How much additional time is required for it to cool to 30°C? 26. A body of unknown temperature is placed in a freezer which is kept at a constant temperature of 0°F. After 15 minutes the temperature of 13 14 Johann Heinrich Lambert (1728–1777) was a Swiss–German astronomer, mathematician, physicist, and man of learning. He was mainly self-educated, and published works on the orbits of comets, the theory of light, and the construction of maps. The Lambert equal-area projection is well known to all cartographers. He is remembered among mathematicians for having given the first proof that π is irrational. Newton himself applied this rule to estimate the temperature of a red-hot iron ball. So little was known about the laws of heat transfer at that time that his result was only a rough approximation, but it was certainly better than nothing. The Nature of Differential Equations 31 the body is 30°F and after 30 minutes it is 15°F. What was the initial temperature of the body? Solve this problem by merely thinking about it, and also by solving a suitable differential equation. 27. A pot of carrot-and-garlic soup cooling in air at 0°C was initially boiling at 100°C and cooled 20° during the first 30 minutes. How much will it cool during the next 30 minutes? 28. For obvious reasons, the dissecting-room of a certain coroner is kept very cool at a constant temperature of 5°C (= 41°F). While doing an autopsy early one morning on a murder victim, the coroner himself is killed and the victim’s body is stolen. At 10 a.m. the coroner’s assistant discovers his chief’s body and finds its temperature to be 23°C, and at noon the body’s temperature is down to 18.5°C. Assuming the coroner had a normal temperature of 37°C (= 98.6°F) when he was alive, when was he murdered?15 29. The radiocarbon in living wood decays at the rate of 15.30 disintegrations per minute (dpm) per gram of contained carbon. Using 5600 years as the half-life of radiocarbon, estimate the age of each of the following specimens discovered by archaeologists and tested for radioactivity in 1950: (a) a piece of a chair leg from the tomb of King Tutankhamen, 10.14 dpm; (b) a piece of a beam of a house built in Babylon during the reign of King Hammurabi, 9.52 dpm; (c) dung of a giant sloth found 6 feet 4 inches under the surface of the ground inside Gypsum C...

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